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The second order Huang-Yang formula to the 3D Fermi gas: the Gross-Pitaevskii regime
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abstract
For a system of $N$ Fermions of spin $1/2$, with its interaction potential of scattering length $a$, the classical Huang-Yang formula states that the energy density $e(\rho)$ is of the form \begin{equation*} e(\rho)=\frac{3}{5}(3\pi^2)^{\frac{2}{3}}\rho^{\frac{5}{3}}+2\pi a\rho^2 +\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}\pi^{\frac{2}{3}}a^2\rho^{\frac{7}{3}} +o(\rho^{\frac{7}{3}}). \end{equation*} We consider a general system of $N$ Fermions of spin $1/\mathbf{q}$ with the scattering length of the interaction potential at the scale $a\propto N^{-1}$, that is, in the Gross-Pitaevskii regime. We prove the 2nd order ground state energy approximation corresponding to the Huang-Yang formula. The thermodynamic limit case shares a similar logic, and could be dealt with in a separate paper.
Forward citations
Cited by 3 Pith papers
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The second order Huang-Yang approximation to the Fermi thermodynamic pressure
The pressure of a dilute Fermi gas is shown to match the Huang-Yang second-order formula up to a temperature that is optimal by scaling, with an explicit positive-temperature correction.
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The Huang-Yang conjecture for the low-density Fermi gas
For dilute spin-1/2 Fermi gases with repulsive short-range interactions, the ground state energy density equals the free Fermi term plus 8πa ρ↑ρ↓ plus the Huang-Yang a²ρ^{7/3} correction, with error o(ρ^{7/3}).
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Semi-classical limit of an attractive Fermi gas in one or two dimensions
For trapped attractive Fermi gases in 1D and 2D, as N grows the ground-state energy approaches the Thomas-Fermi energy, and ground states converge via Husimi functions.
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